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Flat knot 6.1656

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,1,2,0,1,1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1656']
Arrow polynomial of the knot is: 8K13 - 10K1*2 - 8K1K2 - 2K1 + 5K2 + 2K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794']
Outer characteristic polynomial of the knot is: t^7+45t^5+54t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1656']
2-strand cable arrow polynomial of the knot is: -1216K14K2*2 + 2592K1*4K2 - 4448K14 + 992K1*3K2K3 - 640K1*3K3 - 768K12K2*4 + 4256K1*2K2*3 + 64K1*2K2*2K4 - 11808K12K2*2 - 960K1*2K2K4 + 11208K1*2K2 - 224K12K3*2 - 4312K1*2 + 1632K1K23K3 - 2880K1K2*2K3 - 352K1K2*2K5 - 160K1K2K3K4 + 8120K1K2K3 + 800K1K3K4 + 56K1K4K5 - 64K2*6 + 192K2*4K4 - 3128K24 - 64K2*3K6 - 864K22K3*2 - 128K2*2K4*2 + 2520K2*2K4 - 2892K22 + 432K2K3K5 + 80K2K4K6 - 1456K3*2 - 466K4*2 - 64K5*2 - 12K6**2 + 3960
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1656']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4685', 'vk6.4988', 'vk6.6165', 'vk6.6638', 'vk6.8158', 'vk6.8574', 'vk6.9548', 'vk6.9891', 'vk6.20695', 'vk6.22133', 'vk6.28220', 'vk6.29643', 'vk6.39680', 'vk6.41919', 'vk6.46260', 'vk6.47865', 'vk6.48717', 'vk6.48932', 'vk6.49499', 'vk6.49708', 'vk6.50739', 'vk6.50944', 'vk6.51218', 'vk6.51415', 'vk6.57622', 'vk6.58778', 'vk6.62298', 'vk6.63229', 'vk6.67092', 'vk6.67954', 'vk6.69696', 'vk6.70377']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,1,2,0,1,1,-1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1656']

Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794']

Outer characteristic polynomial of the knot is: t^7+45t^5+54t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1656']

2-strand cable arrow polynomial of the knot is: -1216*K1**4*K2**2 + 2592*K1**4*K2 - 4448*K1**4 + 992*K1**3*K2*K3 - 640*K1**3*K3 - 768*K1**2*K2**4 + 4256*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 11808*K1**2*K2**2 - 960*K1**2*K2*K4 + 11208*K1**2*K2 - 224*K1**2*K3**2 - 4312*K1**2 + 1632*K1*K2**3*K3 - 2880*K1*K2**2*K3 - 352*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8120*K1*K2*K3 + 800*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 3128*K2**4 - 64*K2**3*K6 - 864*K2**2*K3**2 - 128*K2**2*K4**2 + 2520*K2**2*K4 - 2892*K2**2 + 432*K2*K3*K5 + 80*K2*K4*K6 - 1456*K3**2 - 466*K4**2 - 64*K5**2 - 12*K6**2 + 3960

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1656']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4685', 'vk6.4988', 'vk6.6165', 'vk6.6638', 'vk6.8158', 'vk6.8574', 'vk6.9548', 'vk6.9891', 'vk6.20695', 'vk6.22133', 'vk6.28220', 'vk6.29643', 'vk6.39680', 'vk6.41919', 'vk6.46260', 'vk6.47865', 'vk6.48717', 'vk6.48932', 'vk6.49499', 'vk6.49708', 'vk6.50739', 'vk6.50944', 'vk6.51218', 'vk6.51415', 'vk6.57622', 'vk6.58778', 'vk6.62298', 'vk6.63229', 'vk6.67092', 'vk6.67954', 'vk6.69696', 'vk6.70377']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is c.

The reverse -K is

The mirror image K* is

The reversed mirror image -K* is

The fillings (up to the first 10) associated to the algebraic genus:

Or click here to check the fillings

invariant	value
Gauss code	O1O2O3U1O4U5U6U3O5O6U4U2
R3 orbit	{'O1O2O3U1O4U5U6U3O5O6U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U4O5O6U1U5U6O4U3
Gauss code of K*	O1O2O3U1U2O4O5U6U5U3O6U4
Gauss code of -K*	O1O2O3U4O5U1U6U5O6O4U2U3
Diagrammatic symmetry type	c
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -2 1 2 1 -2 0],[ 2 0 2 1 1 1 1],[-1 -2 0 1 1 -3 -1],[-2 -1 -1 0 -1 -2 -1],[-1 -1 -1 1 0 -2 0],[ 2 -1 3 2 2 0 1],[ 0 -1 1 1 0 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 -1 -1 -1 -1 -2],[-1 1 0 1 -1 -2 -3],[-1 1 -1 0 0 -1 -2],[ 0 1 1 0 0 -1 -1],[ 2 1 2 1 1 0 1],[ 2 2 3 2 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,1,1,1,1,2,-1,1,2,3,0,1,2,1,1,-1]
Phi over symmetry	[-2,-2,0,1,1,2,-1,1,1,2,3,1,0,1,2,0,1,1,-1,0,0]
Phi of -K	[-2,-2,0,1,1,2,-1,1,1,2,3,1,0,1,2,0,1,1,-1,0,0]
Phi of K*	[-2,-1,-1,0,2,2,0,0,1,2,3,-1,1,1,2,0,0,1,1,1,-1]
Phi of -K*	[-2,-2,0,1,1,2,-1,1,2,3,2,1,1,2,1,0,1,1,-1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+31t^4+33t^2
Outer characteristic polynomial	t^7+45t^5+54t^3+3t
Flat arrow polynomial	8K13 - 10K1*2 - 8K1K2 - 2K1 + 5K2 + 2K3 + 6
2-strand cable arrow polynomial	-1216K14K2*2 + 2592K1*4K2 - 4448K14 + 992K1*3K2K3 - 640K1*3K3 - 768K12K2*4 + 4256K1*2K2*3 + 64K1*2K2*2K4 - 11808K12K2*2 - 960K1*2K2K4 + 11208K1*2K2 - 224K12K3*2 - 4312K1*2 + 1632K1K23K3 - 2880K1K2*2K3 - 352K1K2*2K5 - 160K1K2K3K4 + 8120K1K2K3 + 800K1K3K4 + 56K1K4K5 - 64K2*6 + 192K2*4K4 - 3128K24 - 64K2*3K6 - 864K22K3*2 - 128K2*2K4*2 + 2520K2*2K4 - 2892K22 + 432K2K3K5 + 80K2K4K6 - 1456K3*2 - 466K4*2 - 64K5*2 - 12K6**2 + 3960
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice	False

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